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Abstract Positive lightning leaders are a ubiquitous, yet poorly understood, component of lightning flashes. Upward lightning started by positive leaders may be formed when nearby storm activity induces electrical charges in a tall structure, such as communications towers or wind turbines. Alternatively, upward lightning can be triggered with the rocket‐and‐wire technique. In this paper, we introduce a new self‐consistent model for this important discharge mode, one which solves Maxwell's equations under the quasi‐electrostatic approximation. The model also includes a realistic treatment of the nonlinear plasma conductivity within the leader channel. This new computational tool explains the origin of the positive leader speed, of 10s of km/s, as well as why it displays a steady behavior over time. The model also explains the temporal evolution of current to ground measured during the early stages of rocket‐triggered lightning, where the current exhibits a series of small‐amplitude pulses, which disappear over time. The article also outlines straightforward criteria for leader inception, which may have practical applications for lightning protection.